6 Spatial aspects of the quantum world

In a non-relativistic world, a maximally fuzzy momentum and, consequently, an infinite mean energy would be associated with a sharply localized particle. In a relativistic world, the attempt to produce a strictly localized particle results instead in the creation of particle-antiparticle pairs. It is therefore safe to say that no material object ever has a sharp position (relative to any other object). This implies something of paramount importance: the spatiotemporal differentiation of the physical world is incomplete — it does not go “all the way down.”

To see what exactly this means, let R3(O) be the (imaginary) set of exact positions (labeled by triplets of real numbers) relative to some object O. If no material object ever has a sharp position, we can conceive of a partition of R3(O) into finite regions that are so small that none of them is the sensitive region of an actually existing detector. Hence we can conceive of a partition of R3(O) into sufficiently small but finite regions Rk — k = 1,2,3,… — of which the following is true: there is no object Q and no region Rk such that the proposition “Q is inside Rk” has a truth value. In other words, there is no object Q and no region Rk such that Rk exists for Q. But if a region of space does not exist for any material object, it does not exist at all. The regions Rk — or the distinctions we make between them — correspond to nothing in the actual world. They exist solely in our minds.

What holds for the world’s spatial differentiation holds for its temporal differentiation as well. The times at which observables possess values, like the possessed values themselves, must be indicated in order to exist. Clocks are needed not only to indicate time but also, and in the first place, to make times available for attribution to indicated values. Since clocks indicate times by the positions of their hands, the world’s incomplete temporal differentiation follows from its incomplete spatial differentiation. (Digital clocks indicate times by transitions from one reading to another, without hands. The uncertainty principle for energy and time, however, implies that such a transition cannot occur at an exact time, except in the unphysical limit of infinite mean energy.[1])

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The shapes of things

There is one notion that is decidedly at odds with the incomplete spatial differentiation of the physical world. It is the notion that the “ultimate constituents” of matter are pointlike (or, God help us, stringlike[2]). A fundamental particle — in the context of the Standard Model, a lepton or a quark — is a particle that lacks internal structure. It lacks internal relations; equivalently, it lacks parts. This could mean that it is a pointlike object, but it could also mean that it is formless. (In order to leave a visible trace — a string of bubbles in a bubble chamber, a trail of droplets in a cloud chamber, or something like that — an electron does not need a shape; it only needs to be there. In fact, it was where it was only because its past whereabouts are indicated by bubbles or droplets or some such thing.)

What does the theory have to say on this issue? It obviously favors the latter possibility, inasmuch as nothing in the formalism of quantum mechanics refers to the shape of an object that lacks internal structure.

And experiments? While they can provide evidence of internal structure, they cannot provide evidence of the absence of internal structure. Hence they cannot provide evidence of a pointlike form.

The notion that an object without internal structure has a pointlike form — or any form, for that matter — is therefore unwarranted on both theoretical and experimental grounds.

In addition, it explains nothing. Specifically, it does not explain why a composite object — be it a nucleon, a molecule, or a galaxy — has the shape that it does, inasmuch as all empirically accessible forms are fully accounted for by the relative positions (and orientations) of their material parts. All it does is encumber our efforts to make sense of the quantum world with a type of form whose existence is completely unverifiable, which is explanatorily completely useless, and which differs radically from all empirically accessible forms. If we reject this notion, we arrive at an appealingly uniform concept of form, since then all forms resolve themselves into sets of spatial relations — between parts whose forms are themselves sets of spatial relations, and ultimately between formless parts.

Does the expanse over which these positions are “smeared out” have parts? If it had, the positions themselves would have parts; they would be divided by the parts of space. But this makes no sense. One can divide an object, and thereby create as many positions as there are parts (one for each part) or as many relative positions as there are pairs of parts, but one cannot divide a position. The expanse over which a fuzzy position is probabilistically distributed therefore lacks parts. This confirms an earlier conclusion: if at all we think of space as an expanse, we must think of it as intrinsically undivided.

Instead of attributing the property of spatiality — of being spatially extended — to an expanse to which all relative positions owe their spatial character, we may attribute it to the relative positions themselves. If we do so, there is no need of positing an independently existing expanse; space is nothing but the set of all relations that share this property. If we think this through, we arrive at the following conclusions: Space contains the forms of all things that have forms, for the totality of spatial relations contains — in the proper, set-theoretic sense of containment — the specific sets of spatial relations that constitute material forms. What it does not contain is the corresponding relata — the formless “ultimate constituents” of matter. And if we give the name of “matter” to these “ultimate constituents,” it does not contain matter either.